Entanglement Recovering Under Local Non-Positive Reduced Dynamics

Abstract

Consider a bipartite quantum system \(S=AB\) such that each part interacts only with its local environment. Under such circumstances, one expects that the entanglement between parts A and B does not increase after the time evolution. In fact, this is the case if the reduced dynamics of the system is given by \(\mathcal {E}_{A}\otimes \mathcal {E}_{B}\), where \(\mathcal {E}_{A}\) and \(\mathcal {E}_{B}\) are quantum channels, i.e., completely positive trace-preserving maps. But, the reduced dynamics of the system may be given by a map as \(\Psi _{A}\otimes \Psi _{B}\), where \(\Psi _{A}\) and \(\Psi _{B}\) are local non-positive maps. Then, the entanglement between A and B can exceed its initial value, as was shown in the case studied by Jordan et al. [ Phys. Rev. A 76, 022102 (2007)]. In this paper, we first explore the general circumstances under which one can find such cases as they found. Next, we introduce another general procedure which leads to local non-positive maps that exhibit entanglement exceeding. Furthermore, we demonstrate that this apparent entanglement increase in the two above procedures corresponds physically to the retrieval of pre-existing inaccessible entanglement stored across the global system-environment degrees of freedom, rather than genuine generation of the entanglement ex nihilo.